 # Quick Review: Quadrilaterals and Polygons

A simple closed four sided figure is known as a Quadrilateral.
• Parallelogram: In a parallelogram, opposite sides are parallel and congruent.
Area = length * height
Perimeter = 2 (length + breadth)
• Rectangle: In a rectangle, opposite sides are parallel and congruent.
Diagonal of the rectangle = √(l2 + b2)
Perimeter = 2 (length + breadth)
• Square: All sides and angles of a square are congruent.
Length of the diagonal = l √2 (where l is the length of the side of a square)
Area = (Side)2
Perimeter = 4 X side
• Rhombus: All sides and all opposite angles of a rhombus are congruent. Also, the diagonals are perpendicular to and each other.
Area = (a*b)/2 (where a and b are the lengths of the diagonals of a rhombus)
Perimeter = 4 * length
• Trapezium: No sides, angles and diagonals of a trapezium are parallel to each other.
Area = (1/2)h*(L+L2)
Perimeter = L + L1 + L2 + L3 ###### Important Points
• Of all the quadrilaterals of the same perimeter, the one with the maximum area is the square.
• The quadrilateral formed by joining the midpoints of the sides of any quadrilateral is always a parallelogram (rhombus in case of a rectangle, rectangle in case of a rhombus and square in case of a square).
• The quadrilateral formed by the angle bisectors of the angles of a parallelogram is a rectangle.
• For a rhombus □ABCD, if the diagonals are AC and BD, then AC2 + BD2 = 4 * AB2.
• If a square is formed by joining the midpoints of a square, then the side of the smaller square = side of the bigger square/√2.
• If P is a point inside a rectangle □ABCD, then AP2 + CP2 = BP2 + DP2.
• The segment joining the midpoints of the non-parallel sides of a trapezium is parallel to the two parallel sides and is half the sum of the parallel sides.
• For a trapezium □ABCD, if the diagonals are AC and BD, and AB and CD are the parallel sides, then AC2 + BD2 = AD2 + BC2 + 2 * AB * BD.
• If the length of the sides of a cyclic quadrilaterals area, b, c and d, then it's area = √(s-a)(s-b)(s-c)(s-d) , where s is the semi - perimeter.
• The opposite angles of a cyclic quadrilateral are supplementary.
###### Polygons ###### Properties
• The sides an of regular inscribed polygons, where R is the radius of the circumscribed circle = an = 2R sin 180o/n
• Area of a polygon of perimeter P and radius of in-circle r = (P*r)/2
• The sum of the interior angles of a convex POLYGON, having n sides is 180o (n - 2).
• The sum of the exterior angles of a convex polygon, taken one at each vertex, is 360o.
• The measure of an exterior angle of a regular n- sided polygon is 360o/n .
• The measure of the interior angle of a regular n-sided polygon is (n-2)180o/n
• The number of diagonals of in an n-sided polygon is n(n-3)/2